A homotopy coherent nerve for (∞,n)-categories · arXivDesk
2208.02745Aug 4, 202255 pages; v3: final version to appear in JPAA. v2 supersedes v1, which contains some errors. v2 is a complete rewrite that proves a similar main result to v1 but in the model of Segal category objects in (oo,n-1)-categories instead of complete Segal objects in (oo,n-1)-categories
In the case of (∞,1)-categories, the homotopy coherent nerve gives a right Quillen equivalence between the models of simplicially enriched categories and of quasi-categories. This shows that homotopy coherent diagrams of (∞,1)-categories can equivalently be defined as functors of quasi-categories or as simplicially enriched functors out of the homotopy coherent categorifications. In this paper, we construct a homotopy coherent nerve for (∞,n)-categories. We show that it realizes a right Quillen equivalence between the models of categories strictly enriched in (∞,n−1)
Nearby in the stack
-categories and of Segal category objects in
(∞,n−1)
-categories. This similarly enables us to define homotopy coherent diagrams of
(∞,n)
-categories equivalently as functors of Segal category objects or as strictly enriched functors out of the homotopy coherent categorifications.